Zu Chongzhi

Zu Chongzhi's name is sometimes written as Tsu Ch'ung Chi. He came from a famous family who were originally from Hopeh province in northern China. His great grandfather was an official at the court of the Eastern Chin dynasty which had been established at Jiankang (now Nanking). Weakened by court intrigues, the Eastern Chin dynasty was replaced after a revolt by the Liu-Sung dynasty in 420. Zu Chongzhi's grandfather and father both served as officials of the Liu-Sung dynasty which also had its court at Jiankang (now Nanking).

The Zu family was an extremely talented one with successive generations being, in addition to court officials, astronomers with special interests in the calendar. In ancient China there was a belief that an emperor received his right to rule from heaven. Producng a calendar specifically for a new emperor established a link from the heavens to the particular ruler. This meant that astronomers had important roles at court for their skills could result in an emperor's successful rule. The Zu family handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted.

Zu Chongzhi, in the family tradition, was taught a variety of skills as he grew up. In particular he was taught mathematics, astronomy and the science of the calendar from his talented father. He learnt mathematics from a number of sources, but mainly from Liu Hui's commentary on the Nine Chapters on the Mathematical Art. Zu learnt other skills too for he excelled in engineering and was skilled in literary composition writing ten novels. Zu Chongzhi followed in the family tradition of serving the emperors. He was appointed by the Emperor Xiao-wu (who ruled from 454 to 464) first as an officer in Yang-chou, a city in Kiangsu, and then as an officer in the military staff in Jiankang (now Nanking).

During this time Zu worked on mathematics and astronomy. In particular he was working on a new, more accurate calendar. The calendar which had been in use was based on a 19 year cycle with years consisting of 12 months of 29 or 30 days. In seven of the 19 years an extra month was inserted making it a calendar based both on the sun and the moon with 235 months in 19 years. This had been changed in 412 to a calendar based on a 600 year cycle with an extra month inserted in 221 of the years. This calendar was not accurate enough for Zu.

In 462 Zu proposed a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), to the Emperor which was based on a cycle of 391 years. In 144 of the 391 years an extra month was inserted, so there were 4836 months in 391 years. He was able to make a calendar with this degree of accuracy since he had calculated the length of the tropical year (time between two successive occurrences of the vernal equinox) as 365.24281481 days (an error of only 50 seconds from its true value of 365 days 5 hours 48 minutes 46 seconds), and a nodal month for the moon of 27.21233 days (compare the modern value of 27.21222 days).

Zu, however, had an opponent at the court as far as his calendar was concerned. This was Tai Faxin, one of the Emperor's ministers, who declared that Zu was:-

... distorting the truth about heaven and violating the teaching of the classics.

Zu replied that his calendar was:-

... not from spirits or from ghosts, but from careful observations and accurate mathematical calculations. ... people must be willing to hear and look at proofs in order to understand truth and facts.

Despite having such a powerful opponent as Tai Faxin, Zu won approval for his calendar from Emperor Xiao-wu and the Tam-ing calendar was due to come into use in 464. However, Xiao-wu died in 464 before the calendar was introduced, and his successor was persuaded by Tai Faxin to cancel the introduction of the new calendar. Zu left the imperial service on the death of Emperor Xiao-wu and devoted himself entirely to his scientific studies.

Of course, it is not unreasonable to ask where the numbers 144 and 391 came from. Having accurate knowledge of the lengths of the year and the month were necessary, but it is still not clear how Zu translated this into a cycle of 391 years. In [5] it is suggested that Zu found that there were 365 9589/39491 days in a year and 116321/3939 days in a month. This gives

12 1691772624/4593632611

months in a year. But Zu would know how to reduce fractions to their lowest terms by dividing top and bottom by the greatest common divisor. Doing this gives

1691772624/4593632611 = 144/391

and hence the extra month in 144 out of 391 years.

Before we leave our discussion of Zu's astronomical work we give further details of his work in this area. He was not the first Chinese astronomer to discover the precession of the equinoxes (Yu Xi did so in the fourth century) but he was the first to take this into account in calendar calculations. Because of the precession of the equinoxes the tropical year is shorter by about 21 minutes than the sidereal year (the time taken by the Sun to return to the same place against the background stars). Zu's calculations of the length of the year were well within the range that allowed him to differentiate between the tropical and sidereal year. Jupiter takes about 12 years to complete its orbit but Zu was able to give a much more accurate value than that. He discovered that in 7 cycles of 12 years, Jupiter had completed seven and one twelfth orbits, giving its sidereal period as 11.859 years (accurate to within one part in 4000).

He gave the rational approximation 355/113 to in his text Zhui shu (Method of Interpolation), which is correct to 6 decimal places. He also proved that

3.1415926 < π < 3.1415927

a remarkable result about which it would be nice to have more details. Sadly Zu Chongzhi's book is lost. It is reported in the History of the Sui dynasty, compiled in the 7th century by Li Chunfeng and others, that (see [1] or [3] for a different translation):-

Zu Chongzhi further devised a precise method [of calculating ]. Taking a circle of diameter 10,000,000 chang, he found the circumference of this circle to be less than 31,415,927 chang and greater than 31,415,926 chang. He deduced from these results that the accurate value of the circumference must lie between these two values. Therefore the precise value of the ratio of the circumference of a circle to its diameter is as 355 to 113, and the approximate value is as 22 to 7.

To compute this accuracy for π, Zu must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundereds of square roots, all to 9 decimal place accuracy. Since his book is lost we will never know exactly how he found the rational approximation 355/113 from the decimal approximation. Historians believe, however, that he knew that

if a/b ≤ c/d then a/b ≤ (a+c)/(b+d) ≤ c/d

for any integers a, b, c, d. He then knew that

3 ≤ π ≤ 22/7

so, approximately,

π = 3.1415926 = (3x + 22y)/(x + 7y)

giving y = 16x approximately, so

π = (3x + 22 × 16x)/(x + 7 × 16x) = 355/113.

Martzloff, in [3] or [4], presents another possible way that Zu might have found 355/113 by luck rather than mathematical skill. However, given that Zu's work was considered very difficult and advanced, it is doubtful that it was found by a lucky numerical accident.

In 656, after editing by Li Chunfeng, the treatise Zhui shu (Method of Interpolation) became a text for the Imperial examinations and it became one of The Ten Classics when reprinted in 1084. However, the Zhui shu was too advanced for the students at the Imperial Academy and it was dropped from the syllabus for that reason. This almost certainly explains why the text has not survived, being lost in the early twelfth century.

In the latter part of his life Zu Chongzhi collaborated with his son, Zu Geng (or Zu Xuan), who was also an outstanding mathematician.

Article by:J J O'Connor and E F Robertson

December 2003

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Zu_Chongzhi.html]